Wednesday, June 22, 2011

Tick-Tack-Toe Game

 Tick-Tack-Toe
David and Angela play a game of tick-tack-toe. In this game, the players try to get three circles or three crosses in a row (horizontal, vertical, or diagonal).

A player always tries to win: if a player can place his own symbol (X or O) in a row which already contains two of his own symbols, he will do so.

A player always tries to avoid that his opponent wins: if a player can place his own symbol (X or O) in a row which already contains two of the symbols of his opponent, he will do so.

Of course, the first rule has precedence over the second rule, because the game can be won in this way.

In the game shown on the right, 6 moves have been done. David plays with crosses (X) and Angela plays with circles (0). However, we don't know who started the game.

Who will win this game?

If you're like me, you may be having trouble seeing the image of the board I tried to create. So here's a more basic image:
O | O |
----------
O | X |
----------
X | X |

1. The X will win no matter if he goes first or second because no matter where the O goes the X has another place to win.

2. I think Angela (O) is going to win. The bottom left X has to be the last move, or else O would have previously been forced to block in either the bottom right or top right spot. So, it is O's turn and she can go in the top right corner for the win.

3. Angela.

Since both players have ways to win, the second-to-last move must have been one that create two possible victories for that player where no victories had been possible before. Angela's marks are the only one for which this is possible, since before the upper left corner was marked with an O, the other two O's were not connected. On the other hand, if you were to remove any one of the X's, it would result in 2 X's in a row. Thus, X must have started, and O played the second-to-last move which forced X to only block one of two possible victories.

4. Angela will win, but the way they have proceeded with the game defies logic.

5. The player who marks next wins. X didn't mark first because if it did, O would make the last mark, and would have been forced to respond to either lowerleft-middle or lowerleft-lowermiddle, which it didn't. There are an even number of markers on the board, and the player who marked first marks next. O wins.

6. I think David only because the directions keeps using the term his and he...lol I no it seems dumb, but o well that's my thinking.

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